COMPRESSED DECISION PROBLEMS FOR GRAPH PRODUCTS AND APPLICATIONS TO (OUTER) AUTOMORPHISM GROUPS

2012 ◽  
Vol 22 (08) ◽  
pp. 1240007 ◽  
Author(s):  
NIKO HAUBOLD ◽  
MARKUS LOHREY ◽  
CHRISTIAN MATHISSEN
Keyword(s):  
Automorphism Group ◽  
Word Problem ◽  
Polynomial Time ◽  
Coxeter Group ◽  
Automorphism Groups ◽  
Outer Automorphism ◽  
Conjugacy Problem ◽  
Artin Group ◽  
Graph Products ◽  
Direct Corollary

It is shown that the compressed word problem of a graph product of finitely generated groups is polynomial time Turing-reducible to the compressed word problems of the vertex groups. A direct corollary of this result is that the word problem for the automorphism group of a right-angled Artin group or a right-angled Coxeter group can be solved in polynomial time. Moreover, it is shown that a restricted variant of the simultaneous compressed conjugacy problem is polynomial time Turing-reducible to the same problem for the vertex groups. A direct corollary of this result is that the word problem for the outer automorphism group of a right-angled Artin group or a right-angled Coxeter group can be solved in polynomial time. Finally, it is shown that the compressed variant of the ordinary conjugacy problem can be solved in polynomial time for right-angled Artin groups.

scholarly journals ON SOLVABLE SUBGROUPS OF AUTOMORPHISM GROUPS OF RIGHT-ANGLED ARTIN GROUPS

2011 ◽  
Vol 21 (01n02) ◽  
pp. 61-70 ◽  
Author(s):  
MATTHEW B. DAY
Keyword(s):  
Automorphism Group ◽  
Finite Index ◽  
Automorphism Groups ◽  
Outer Automorphism ◽  
Free Subgroup ◽  
Artin Group ◽  
Artin Groups ◽  
Alternative Theorem ◽  
Right Angled Artin Groups ◽  
Defining Graph

For any right-angled Artin group, we show that its outer automorphism group contains either a finite-index nilpotent subgroup or a nonabelian free subgroup. This is a weak Tits alternative theorem. We find a criterion on the defining graph that determines which case holds. We also consider some examples of solvable subgroups, including one that is not virtually nilpotent and is embedded in a non-obvious way.

Finite Groups Which are Automorphism Groups of Infinite Groups Only

1985 ◽  
Vol 28 (1) ◽  
pp. 84-90
Author(s):  
Jay Zimmerman
Keyword(s):  
Finite Group ◽  
Automorphism Group ◽  
Finite Field ◽  
Finite Groups ◽  
Finite Simple Group ◽  
Automorphism Groups ◽  
Outer Automorphism ◽  
Schur Multiplier ◽  
Infinite Groups ◽  
Infinite Set

AbstractThe object of this paper is to exhibit an infinite set of finite semisimple groups H, each of which is the automorphism group of some infinite group, but of no finite group. We begin the construction by choosing a finite simple group S whose outer automorphism group and Schur multiplier possess certain specified properties. The group H is a certain subgroup of Aut S which contains S. For example, most of the PSL's over a non-prime finite field are candidates for S, and in this case, H is generated by all of the inner, diagonal and graph automorphisms of S.

scholarly journals COMPRESSED WORDS AND AUTOMORPHISMS IN FULLY RESIDUALLY FREE GROUPS

2010 ◽  
Vol 20 (03) ◽  
pp. 343-355 ◽  
Author(s):  
JEREMY MACDONALD
Keyword(s):  
Automorphism Group ◽  
Word Problem ◽  
Free Group ◽  
Polynomial Time ◽  
Free Groups ◽  
Finitely Generated

We show that the compressed word problem in a finitely generated fully residually free group ([Formula: see text]-group) is decidable in polynomial time, and use this result to show that the word problem in the automorphism group of an [Formula: see text]-group is decidable in polynomial time.

Recovering ordered structures from quotients of their automorphism groups

2003 ◽  
Vol 68 (4) ◽  
pp. 1189-1198
Author(s):  
M. Giraudet ◽  
J. K. Truss
Keyword(s):  
Automorphism Group ◽  
Automorphism Groups ◽  
Outer Automorphism ◽  
Lattice Ordered Group ◽  
Ordered Group ◽  
Ordered Structures ◽  
Outer Automorphism Group ◽  
Simple Lattice

AbstractWe show that the ‘tail’ of a doubly homogeneous chain of countable cofinality can be recognized in the quotient of its automorphism group by the subgroup consisting of those elements whose support is bounded above. This extends the authors' earlier result establishing this for the rationals and reals. We deduce that any group is isomorphic to the outer automorphism group of some simple lattice-ordered group.

ALL GROUPS ARE OUTER AUTOMORPHISM GROUPS OF SIMPLE GROUPS

2001 ◽  
Vol 64 (3) ◽  
pp. 565-575 ◽  
Author(s):  
MANFRED DROSTE ◽  
MICHÈLE GIRAUDET ◽  
RÜDIGER GÖBEL
Keyword(s):  
Automorphism Group ◽  
Recent Result ◽  
Automorphism Groups ◽  
Outer Automorphism ◽  
Simple Groups ◽  
Symmetry Groups ◽  
Normal Subgroups ◽  
Outer Automorphism Group ◽  
Relational Structures ◽  
Outer Automorphism Groups

It is shown that each group is the outer automorphism group of a simple group. Surprisingly, the proof is mainly based on the theory of ordered or relational structures and their symmetry groups. By a recent result of Droste and Shelah, any group is the outer automorphism group Out (Aut T) of the automorphism group Aut T of a doubly homogeneous chain (T, [les ]). However, Aut T is never simple. Following recent investigations on automorphism groups of circles, it is possible to turn (T, [les ]) into a circle C such that Out (Aut T) [bcong ] Out (Aut C). The unavoidable normal subgroups in Aut T evaporate in Aut C, which is now simple, and the result follows.

scholarly journals Geometric realizations of abstract regular polyhedra with automorphism group H 3

2020 ◽  
Vol 76 (3) ◽  
pp. 358-368
Author(s):  
Jonn Angel L. Aranas ◽  
Mark L. Loyola
Keyword(s):  
Automorphism Group ◽  
Coxeter Group ◽  
Automorphism Groups ◽  
Full Rank ◽  
Geometric Realization ◽  
Regular Polyhedron ◽  
Orthogonal Representation ◽  
Regular Polyhedra ◽  
Open Set

A geometric realization of an abstract polyhedron {\cal P} is a mapping that sends an i-face to an open set of dimension i. This work adapts a method based on Wythoff construction to generate a full rank realization of an abstract regular polyhedron from its automorphism group Γ. The method entails finding a real orthogonal representation of Γ of degree 3 and applying its image to suitably chosen (not necessarily connected) open sets in space. To demonstrate the use of the method, it is applied to the abstract polyhedra whose automorphism groups are isomorphic to the non-crystallographic Coxeter group H 3.

scholarly journals Conjugacy and dynamics in almost automorphism groups of trees

2021 ◽  
pp. 1-49
Author(s):  
Gil Goffer ◽  
Waltraud Lederle
Keyword(s):  
Automorphism Group ◽  
Automorphism Groups ◽  
Conjugacy Problem ◽  
Conjugacy Classes ◽  
Homogeneous Tree ◽  
Regular Tree

We determine when two almost automorphisms of a regular tree are conjugate. This is done by combining the classification of conjugacy classes in the automorphism group of a level-homogeneous tree by Gawron, Nekrashevych and Sushchansky and the solution of the conjugacy problem in Thompson’s [Formula: see text] by Belk and Matucci. We also analyze the dynamics of a tree almost automorphism as a homeomorphism of the boundary of the tree.

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